| L(s) = 1 | + (0.931 − 2.13i)2-s + (−0.702 + 1.58i)3-s + (−2.33 − 2.51i)4-s + (−2.63 − 0.396i)5-s + (2.72 + 2.97i)6-s + (−2.74 − 2.54i)7-s + (−3.14 + 1.10i)8-s + (−2.01 − 2.22i)9-s + (−3.30 + 5.25i)10-s + (−0.317 + 0.273i)11-s + (5.62 − 1.92i)12-s + (0.316 − 0.463i)13-s + (−8.00 + 3.49i)14-s + (2.47 − 3.89i)15-s + (−0.0678 + 0.905i)16-s + (2.10 + 2.10i)17-s + ⋯ |
| L(s) = 1 | + (0.658 − 1.51i)2-s + (−0.405 + 0.914i)3-s + (−1.16 − 1.25i)4-s + (−1.17 − 0.177i)5-s + (1.11 + 1.21i)6-s + (−1.03 − 0.963i)7-s + (−1.11 + 0.389i)8-s + (−0.671 − 0.741i)9-s + (−1.04 + 1.66i)10-s + (−0.0957 + 0.0823i)11-s + (1.62 − 0.556i)12-s + (0.0876 − 0.128i)13-s + (−2.13 + 0.933i)14-s + (0.639 − 1.00i)15-s + (−0.0169 + 0.226i)16-s + (0.510 + 0.510i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.108577 + 0.654373i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.108577 + 0.654373i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.702 - 1.58i)T \) |
| 29 | \( 1 + (4.54 - 2.89i)T \) |
| good | 2 | \( 1 + (-0.931 + 2.13i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (2.63 + 0.396i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (2.74 + 2.54i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (0.317 - 0.273i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.316 + 0.463i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-2.10 - 2.10i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.27 + 2.05i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-0.386 - 0.151i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-8.11 + 5.98i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (0.872 + 2.49i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-8.79 - 2.35i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.97 + 9.45i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (-3.52 - 4.09i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (10.7 + 8.53i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (4.46 - 2.57i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (13.3 - 0.498i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (2.39 - 0.179i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (7.83 - 3.77i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.882 - 7.83i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 + 6.29i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-2.83 + 9.18i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-3.90 + 0.439i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-2.11 - 3.99i)T + (-54.6 + 80.1i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.35295248250585998544743962437, −10.72059688651708062665447368581, −10.02473612370972472141027761591, −9.082888488719127099561015217756, −7.55524019341098121118082908606, −6.00725751922526324696062725342, −4.48381331478217180868784626174, −3.96901395819251044884403170707, −3.08192736482299470200928147044, −0.43484746904673498366057128971,
3.06226040543416437482909488915, 4.54364912158604801411621146738, 5.87917214379845188493501460922, 6.39489818888849580049966599527, 7.48268124929981933518501253273, 8.055107805780699948304251527925, 9.138973290785703953932004201651, 10.90256674466380061330815423029, 12.15477858828143007506739075333, 12.46479714681660300843903165303
